The aperture for UHE tau neutrinos of the Auger fluorescence detector using a Digital Elevation Map

0.78 MB·English

by  Gennaro Miele

DSF-17/2005, IFIC/05-36
The aperture for UHE tau neutrinos
of the Auger fluorescence detector
using a Digital Elevation Map
6 Gennaro Mielea, Sergio Pastorb, Ofelia Pisantia
0
0
a
2 Dipartimento di Scienze Fisiche, Universita` di Napoli “Federico II” and INFN
n Sezione di Napoli, Complesso Universitario di Monte S. Angelo, Via Cinthia,
a I-80126 Napoli, Italy
J
bInstituto de F´ısica Corpuscular (CSIC-Universitat de Val`encia),
1
3 Ed. Institutos de Investigacio´n, Apdo. 22085, E-46071 Valencia, Spain

2
v
8
3 Abstract
0
8
0 We perform a new study of the chances of the fluorescence detector (FD) at the
5 Pierre Auger Observatory to detect the tau leptons produced by Earth-skimming
0 ultrahighenergyν ’s.Wepresentanewandmoredetailedevaluationoftheeffective
/ τ
h apertureof the FD that considers a reliable fiducial volume for the experimental set
p
up.Inaddition,wetakeintoaccounttherealelevationprofileoftheareanearAuger.
-
o
We find a significant increase in the number of expected events with respect to the
r
t predictions of a previous semi-analytical determination, and our results show the
s
a enhancement effect for neutrino detection from the presence of the near mountains.
:
v
i
X
Key words:
r PACS: 95.85.Ry, 13.15.+g, 96.40.Tv, 95.55.Vj, 13.35.Dx
a
Neutrinosconstituteoneofthecomponentsofthecosmicradiationintheultra
high energy (UHE) regime. Since we have detected ultra high energy cosmic
rays(UHECR),thepresenceofasecondaryUHEneutrinofluxisguaranteedas
a result of the π-photoproduction, due to the interaction of hadronic UHECR
with the cosmic microwave background. The detection of these cosmogenic
neutrinos [1,2], in addition to a possible primary neutrino flux, would provide
precious information on the physics and position of their powerful astrophys-
ical sources. On the other hand, copious neutrino fluxes are also predicted
in more exotic top-down scenarios where relic massive particles, produced at
the first moments of the Universe, decay into UHE lighter particles, among
which neutrinos and photons are expected. In any case, the detection of UHE
Preprint submitted to Elsevier Science 5 February 2008
neutrinos would significantly contribute to unveiling the still unknown origin
of UHECR.
Due to the very low expected fluxes and the small neutrino-nucleon cross sec-
18
tion, neutrinos with energies of the order of 10 eV and larger are hardly
detectable even in the new generation of giant array detectors for cosmic radi-
ation, like the Pierre Auger Observatory (Auger, in short) [3,4]. The detection
of UHE neutrinos inducing inclined air showers was recently reviewed in [5].
In particular, a promising strategy concerning the detection of the tau leptons
produced by Earth-skimming UHE ν ’s has been analyzed in a series of papers
τ
18−21
[6]–[19]. UHE τ’s, with energies in the range 10 eV, have a decay length
not much larger than the corresponding interaction range. Thus, if a UHE ν
τ
crosses the Earth almost horizontally (Earth-skimming) and interacts in the
rock, the produced τ has a chance to emerge from the surface and decay in
the atmosphere, producing a shower that in principle can be detected as an
up-going or almost horizontal event.
The aim of this letter is to perform a new, more refined, estimate of the effec-
tive aperture of the fluorescence detector (FD) at Auger to Earth-skimming
UHE ν ’s. A calculation of the number of possible up-going τ showers de-
τ
tectable with the FD has already been performed in ref. [15] by using a semi-
analytical computation. Our analysis represents a considerable improvement
with respect to the estimate of this last work, since it uses a different method
for calculating the number of ν /τ events, which now includes a class of tracks
τ
neglected in the previous calculation.
An additional improvement in our analysis comes from considering the effects
ofthetopologyaroundtheAugerobservatorysite,byusingaDigitalElevation
Map (DEM) of the area around the experiment. A detailed DEM of the Earth
surface is provided by ASTER (Advanced Spaceborne Thermal Emission and
Reflection Radiometer) [20] which is an imaging instrument that is flying
on Terra, a satellite launched in December 1999 as part of NASA’s Earth
Observing System. The available elevation map, GTOPO30, is a global digital
model where the elevations are regularly spaced at 30-arc seconds. We show in
Fig. 1 a 3D map of the relevant region around Auger. We will use this DEM to
produce a realistic and statistically significant sample of possible ν /τ tracks
τ
crossing the fiducial volume of Auger, that will be used later to evaluate the
real aperture of FD at Auger.
We will define the Auger fiducial volume as that limited by the the six lateral
surfaces Σ (the subindex a = W, E, N, S, U and D labels each surface
a
through its orientation: West, East, North, South, Up, and Down), and with
Ω ≡ (θ ,φ ) a generic direction of a track entering Σ , as shown in Fig. 2.
a a a a
We have considered a simplified Auger area, given by a 50×60 km rectangle
(an approximation to the real one, see ref. [4]), while the height of the fiducial
2
Fig. 1. A 3D map in longitude and latitude of the area around Auger with the
elevation (not to scale) expressed in meters. The Auger position and surface, ap-
proximated to a rectangle, is indicated in red.
Fig. 2. A simplified scheme of the Auger fiducial volume is represented (height not
to scale). The lateral surfaces are labelled by their orientation. Two examples of
entering tracks are also shown.
volume was fixed to 10 km in order to be within the range of detection of the
FD eyes. This is a conservative estimate, since we expect that the effective
fiducial volume for the detection at the FD will be larger.
Let Φ be an isotropic flux of ν +ν . By generalizing the formalism developed
ν τ τ
3
in ref. [13], the number of τ leptons emerging from the Earth surface with
energy E , going through Σ and showering in the fiducial volume per unit of
τ a
time (thus potentially detectable by the FD), is given by
dN dΦ (E )
τ ν ν
=D dΩ dS dE
a a ν
dt !a Z Z Z dEνdΩa
× dE ǫ(E ) cos(θ ) k (E ,E ;~r ,Ω ) , (1)
τ τ a a ν τ a a
Z
where D is the duty cycle and ǫ(E ) is the detection efficiency of the FD,
τ
18
respectively [3]. The minimum energy for the τ leptons, 10 eV, is chosen
taking into account the energy threshold for the flourescence process [15]. The
kernel k (E ,E ;~r ,Ω ) is theprobability that anincoming neutrino crossing
a ν τ a a
the Earth with energy E and direction Ω , produces a lepton emerging with
ν a
energy E , which enters the fiducial volume through the lateral surface dS
τ a
at the position ~r and decays inside this volume (see Fig. 2 for the angle
a
definition). In Eq. (1), due to the very high energy of ν , we can assume that
τ
in the process ν + N → τ + X the charged lepton is produced along the
τ
neutrino direction.
As already shown in details in ref. [15], this process can occur if the following
conditions are fulfilled,
1) the ν with energy E has to survive along a distance z through the Earth;
τ ν
2) the neutrino converts into a τ in the interval z,z +dz;
3) the created τ emerges from the Earth before decaying with energy E ;
τ
4) the τ lepton enters the fiducial volume through the lateral surface Σ at the
a
point ~r and decays inside this volume.
a
1) The probability P1 that a neutrino with energy Eν crossing the Earth
survives up to a certain distance z inside the rock is
z
P1 = exp − , (2)
( λνCC(Eν))
where
1
λν (E ) = (3)
CC ν σνN(E )̺ N
CC ν s A
3
is the charged current (CC) interaction length in rock (̺ ≃ 2.65 g/cm ).
s
A detailed discussion of an updated evaluation of the neutrino-nucleon cross
section, σνN(E ), can be found in ref. [15]. The dependence of λν on the
CC ν CC
particular track direction can be safely neglected, since it is well known that
4
the interesting events are almost horizontal and thus the experienced Earth
1
density is essentially equal to ̺ .
s
2) The probability for ν → τ conversion in the interval [z, z +dz] is
τ
dz
P2dz = . (4)
λν (E )
CC ν
3) The probability P that a charged lepton survives as it loses its energy
c
travelling through the Earth is described by the coupled differential equations
dP m
c τ
=− P , (5)
c
dz cτ E
τ τ
dE
τ
=−(β +γ E ) E ̺ . (6)
τ τ τ τ s
dz
−13
Here m = 1.77 GeV, τ ≃ 3.4×10 s denotes the τ mean lifetime, whereas
τ τ
−6 2 −1 −18 2 −1
the parameters β ≃ 0.71 × 10 cm g and γ ≃ 0.35 × 10 cm g
τ τ
−1
GeV , as discussed in ref. [15], fairly describe the τ energy loss in matter (for
further references and a recent discussion, see [21]). We denote the transferred
0 0
energy as E = E (E ) = (1−hy i)E (see ref. [15] for details). By solving
τ τ ν CC ν
Eq.s (5) and (6) at the emerging point on the Earth surface one has
m 1 1
ω τ
P =(F(E ,E )) exp − − , (7)
c ν τ ( cττβτ̺s Eτ Eτ0(Eν)!)
0
E = βτ Eτ(Eν) exp{−̺sβτ(zmax −z)} , (8)
τ βτ +γτ Eτ0(Eν)(1−exp{−̺sβτ(zmax −z)})
where
0
E (E )(β +γ E ) m γ
F(E ,E ) ≡ τ ν τ τ τ , ω ≡ τ τ . (9)
ν τ E (β +γ E0(E )) cτ β2̺
τ τ τ τ ν τ τ s
The quantity zmax = zmax(~ra,Ωa) represents the total length in rock for a
given track entering the lateral surface Σ of the fiducial volume at the point
a
~r and with direction Ω .
a a
The energy E of the exiting lepton must be consistent with Eq. (8). This
τ
condition is enforced by the presence of a δ-function in the final expression of
1
WeneglectthefactthatUHEneutrinosmaytraverseasignificantamountofwater
(as those from the West), which could enhance the arrival of UHE tau neutrinos at
the largest energies, see e.g. [14].
5
the probability P3,
0
P3 = Pc δ Eτ − βτ +γβττEEτ0τ((EEνν))(e1x−p{e−xp̺s{β−τ̺(szmβτax(z−mazx)−} z)})! . (10)
4) Once the τ lepton has emerged from the Earth surface, its showering prob-
ability is determined by the total distance it has to travel for reaching the
fiducial volume. If we denote with λout = λout(~ra,Ωa) the total length trav-
elled in the atmosphere by the τ before reaching the point ~r on the surface
a
Σa, and with λin = λin(~ra,Ωa) the length of the intersection of the track with
the fiducial volume, the decay probability inside the fiducial volume is given
by
λoutmτ λinmτ
P4 = exp − 1−exp − . (11)
( cττ Eτ ) ( cττ Eτ)!
Collecting together the different probabilities in Eq.s (2), (4), (10) and (11),
we have
zmax
ka(Eν,Eτ ;~ra,Ωa) = P1P2P3P4 dz . (12)
Z0
Since the flux can be reasonably assumed isotropic, namely
dΦ (E ) dΦ (E )
ν ν ν ν
= ∀a , (13)
dE dΩ dE dΩ
ν a ν
we can rewrite Eq. (1), summing over all the surfaces, as
dN dΦ (E )
τ ν ν
= D dE A(E ) , (14)
ν ν
dt dE dΩ
Z ν
where the effective aperture, A(E ) ≡ A (E ), is the sum of each surface
ν a a ν
contribution,
P
A (E ) ≡ dE K (E ,E ) , (15)
a ν τ a ν τ
Z
and
K (E ,E )= dΩ dS cos(θ ) ǫ(E )k (E ,E ;~r ,Ω )
a ν τ a a a τ a ν τ a a
Z Z
6
Fig. 3. The total effective aperture A(E ) is plotted versus the neutrino energy
ν
(solid line). The dashed line corresponds to the same quantity as obtained in ref.
[15] for H = 30 km.
zmax
= dΩa dSa cos(θa) ǫ(Eτ) P1P2P3P4 dz . (16)
Z Z Z0
In order to get the explicit expression for K (E ,E ) an extremely involved
a ν τ
integration has to be performed, which requires the computation of all the
properties for each track (~r , Ω ), taking into account the DEM of the Auger
a a
site. To this aim a suitable approach is based on the following procedure:
we use the available DEM of the Auger area to isotropically generate a large
number oforientedtracks(letussayN)whichcrosstheAugerfiducialvolume.
If we denote with N the subset of the N tracks which enter through surface
a
Σ , then the kernel K (E ,E ) can be well approximated by the expression
a a ν τ
S Na
a
K (E ,E ) ≈ 2πǫ(E ) cos(θ ) k (E ,E ;~r ,Ω ) . (17)
a ν τ τ N ia a ν τ ia ia
a iXa=1
We show in Fig. 3 the aperture A(E ) as a function of the neutrino energy
ν
(solid line), compared with the results of ref. [15] (dashed line). Remarkably,
an enhancement factor approximately of one order of magnitude is found with
respect to the semi-analytical results of ref. [15]. The origin of the difference
between the two calculations can be explained in the following way. In ref. [15]
an event is rejected if the τ takes more than 30 km to decay; this means that
almost horizontal events (where the τ emerges from the Earth far from the
lower surfaceoftheexperiment, travels morethan30km beforeentering inthe
7
Fig. 4. The effective apertures A (E ) defined in Eq. (15) are plotted versus the
a ν
neutrino energy. The thin solid line corresponds to the same quantity as obtained
in ref. [15] for H =30 km.
fiducial volume, and then decays inside it), preferred by the lateral surfaces
due to the cos(θ ) term in Eq. (17), were not included in the calculation
ia
of ref. [15], while they are present in this analysis. In Fig. 4 the different
contributions from each surface are plotted together with the results of ref.
[15] (thin solid line). As one can notice, the contribution coming from the
tracks going through the Down surface (thick solid line) is in nice agreement
with our previous calculation. Moreover, it is possible to distinguish a North-
South effect and, more consistent, a West-East effect.
It is worth observing the different high energy behavior of the aperture cor-
responding to the D surface, A , with respect to the others. The increase of
D
the neutrino energy would select the almost horizontal tracks, which however
are depressed, for the D-surface only with respect to the lateral ones, by the
factor cos(θ ).
iD
We can use the expression in Eq. (14) to obtain the yearly number of τ show-
ering events at the FD of Auger (assuming a duty cycle D = 10%). In Table
1 these rates are reported for the same UHE neutrino fluxes considered in
section 2 of ref. [15] (see in particular figs. 1 and 2), and described in a series
of papers [22,23,24,25,26,27,28]. The three GZK fluxes refer to three possible
scenarios for cosmogenic neutrinos, which are those produced from an initial
flux of UHE protons. Instead the NH (New Hadrons) and TD (Topological
Defects) cases are two examples of exotic models capable of generating the
10
UHECR above 10 eV, with large associated neutrino fluxes. For each neu-
trino flux (fixed column), we list the total number of yearly events as well as
8
GZK-WB GZK-L GZK-H NH TD
Surface D 0.016 0.040 0.095 0.246 0.100
Surface S 0.012 0.037 0.098 0.214 0.094
Surface N 0.015 0.046 0.125 0.267 0.120
Surface W 0.022 0.066 0.181 0.380 0.174
Surface E 0.008 0.024 0.061 0.139 0.060
Total 0.074 0.213 0.560 1.245 0.548
Ref. [15] 0.02 0.04 0.09 0.25 0.11
Table 1
Yearly rate of Earth-skimming events at the FD for the different neutrino fluxes
considered in ref. [15]. The number of τ’s showering into the fiducial volume that
enter througheach lateral surfacearereported,aswell asthetotal numberofevents
for each flux. For comparison, we include the corresponding results from ref. [15].
the contributions from each lateral surface.
Some comments arein turn.The number of events crossing thebottomsurface
isinfairagreement withthepreviousanalyticalresultofref.[15].However, the
total number of events is a factor 4-6 larger (depending on the model consid-
ered), showing that the main contribution to the number of events is coming
from almost horizontal showers, where the τ emerges from Earth surface far
away from the Auger fiducial volume and decays inside it. The enhancement
factor depends on the different features of the fluxes used in the analysis. For
example, for the three GZK models in Table 1, this factor ranges from 3.7 to
6.2, corresponding to a hardening of the differential fluxes in energy (see Figs.
1 and 2 of ref. [15]). Instead, the enhancement is roughly the same for the last
two models in Table 1 despite the fact that they have a different spectrum in
the high energy range, due to the suppression of the very high energy neutrino
events which escape without showering.
AsonecanseefromTable1,asignificant differenceinthenumber ofevents ex-
istsbetweentheSurfacesW(facingtheAndes)andE,whichshowsamountain
effect. This enhancement is mainly due to the largest amount of rock encoun-
teredbyhorizontaltracks comingfromthewest side. Ifwedefine,asameasure
of the effect of mountains, the difference of the number of expected events en-
tering the volume from W-surface and E-surface, divided by their average, the
effect can be quite remarkable (even order 30%). This effect would also be
larger by comparing the exclusive apertures, but in this case the difference in
the apertures, which is larger for larger neutrino energy, should compel the
fast decreasing with energy of neutrino fluxes. Of course, on the total number
of events this 30% effect is diluted because it concerns only one surface among
four lateral ones. Moreover, a similar but smaller effect (of order ∼ 20-25%) is
9
GZK-WB GZK-L GZK-H NH TD
ν 0.034 0.098 0.277 0.565 0.276
e
ν 0.009 0.018 0.036 0.113 0.043
τ
Total 0.043 0.116 0.313 0.678 0.319
Table 2
Yearly expected number of down-going events at the FD, due to the showering of
ν and ν inside the Auger fiducial volume. The different predictions refer to the
e τ
◦
same fluxes of Table 1 and to a zenith angle larger than 60 .
also present in the difference between the number of events from the Surfaces
N (facing the higher part of the cordillera) and S.
In conclusion, the largest contribution through the surfaces E, W, S and N
makesthetotalnumber ofyearlyEarth-skimming events largerthantheprevi-
ous estimates of ref. [15]. This in turn increases the detection chances of UHE
ν at the FD of Auger, which seem realistic even for conservative neutrino
τ
fluxes like GZK-WB, considering that data will be taken over many years of
observation.
In analogy with the analysis of ref. [15], we report in Table 2 the expected
numbers of yearly down-going events at FD, produced by UHE ν and ν
e τ
2
showering inside thefiducial volume .Thenumbers arecalculated ina similar
way to the Earth-skimming case, but with a different kernel in Eq. (12),
k (E ,E ;~r ,Ω ) = δ E −E0(E ) Pνα Pνα (18)
a να α a a α α να 1 2
(cid:16) (cid:17)
0
where α = e,τ and E (E ) is the transferred energy to the charged lepton.
α να
For a ν we have
e
Pνe=exp − λout
1 ( λνCC(Eνe))
Pνe= λindz exp{−z/λνCC(Eνe)} = 1−exp − λin , (19)
2 Z0 λνCC(Eνe) ( λνCC(Eνe))
which take into account the probability that the neutrino survives until the
fiducial volume producing an electron which initiates a shower inside the de-
tector. Instead, for a ν we have
τ
2 From the study of ν ,ν –induced shower in the atmosphere of ref.[29] it is argued
e µ
that high energy ν ’s have a very small probability of inducing a shower before
µ
reaching the ground.
10